804 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
84. The only case of importance in this investigation is that in which the tem¬ 
perature varies along a solid surface and is constant at right angles. 
Taking z =—c as the equation to the solid surface, and supposing the gas uniform 
in the direction y, and that l — = ~ = 0, then if xyz are the coordinates of a point P 
and z is greater than —c the effect of the solid surface will he to alter the values 
of s in the terms involving the differentials of p and a. Using a suffix to indicate 
that the values of s for such terms is arbitrary, we may proceed to determine the 
values of cr(Q), as in Art. 82. The important cases are Q = Mm and Q=M. 
Remembering that s L refers to such terms in A C E G as involve ~ or and that 
CL’Jb CVjC 
W= 0, we have by the method of Art. 82 
S —S 1 dpa 2 
2tt dx 
s dU 
ox(M) = fpU 
a-,(M) = lpU 
dpa sp dU 1 
dx 2 dz 1^ 
dpa sp dU | 
dx' 2 dz J 
(53) 
Further, to adapt these equations to the form required, put a l and u 2 for the mean 
w+ y j W— y 
velocity of the opposite groups w-\- and w —, so that cr. r (M)= p- 1 , cr x {M)=p-^. 
Then since u may be taken as constant in the direction of x, we have by corol¬ 
lary to theorem (II.) and equation (51) 
u, u 0 du . 
P~P^ = ~ S P~dz .( 53a ) 
Subtracting equations (53) 
du s—s x dpa dU 
~ Sp lh = 2^r ~dx~ Sp Hz’ 
and substituting for C ~ in equation (52) 
r 2 (Mli): 
S Si 
pOL 
da. 
dx 
s 
\Ztt 
du 
P a Tz 
(54) 
If the point P lies between two surfaces, then putting s. 2 as an arbitrary function we 
have 
/nr \ S-, da S du 
°-.(M M )=— pa ---^ pa ~ 
.(54a) 
For further consideration of s . 2 ~see Art. 96. 
[This section (VII.) was revised and somewhat enlarged in August, 1879, in accord¬ 
ance with a suggestion made by one of the Referees.] 
